They explain that you need xcc.jar, xdbc.jar. and xdmp.jar driver files. I can only find xcc.jar, which I have installed.

When I open the connection Oxygen says the server failed to respond with a valid HTTP response. I'm not a developer and only have experience querying our db through cq, so I can't easily troubleshoot from the server end...

Note that the MarkLogic server has three types (at least) of application servers that can be created and managed from the server management and administration page: HTTP, XDBC and WebDAV

For the XDBC host and port in the Oxygen connection make sure you use the port of an XDBC application server(not HTTP server).For the WebDAV URL make sure you are using the URL of a WebDAV application server. e.g. http://hostname:port